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Example 1 - 10 Coin Flips

I've a coin and my null hypothesis is that it's balanced - which means it has a 0.5 chance of landing heads up. I flip my coin 10 times, which may result in 0 through 10 heads landing up. The probabilities for these outcomes -assuming my coin is really balanced- are shown below.
*
Technically, this is a
binomial distribution
. The formula for computing these probabilities is based on mathematics and the (very general) assumption of independent and identically distributed variables .
Keep in mind that probabilities are relative frequencies. So the 0.24 probability of finding 5 heads means that if I'd draw a 1,000 samples of 10 coin flips, some 24% of those samples should result in 5 heads up.

Now, 9 of my 10 coin flips actually land heads up. The previous figure says that the probability of finding 9 or more heads in a sample of 10 coin flips,
p = 0.01
. If my coin is really balanced, the probability is only 1 in 100 of finding what I just found. So, based on my sample of N = 10 coin flips, I
reject the null hypothesis
: I no longer believe that my coin was balanced after all.

Note that females scored 3.5 points higher than males in this sample. However, samples typically differ somewhat from populations. The question is:
if the mean scores for
all
males and
all
females are equal, then
what's the probability of finding this mean difference
or a more extreme one in a sample of N = 360?
This question is answered by running an
MTNG Women Sports Shoes 69883 C28832 MESH Burdeos Yew9h7W
.

what's the probability of finding this mean difference

So what sample mean differences can we reasonably expect? Well, this depends on

Mathematics is known for its resolute commitment to precision in definitions and statements. However, when words are pulled from the English language and given rigid mathematical definitions, the connotations and colloquial use outside of mathematics still remain. This can lead to immutable mathematical termsbeing used interchangeably, even though the mathematical definitions are not equivalent. This occurs frequently in probability and statistics, particularly with the notion of
uncorrelated
and
independent
. We will focus this post on the exact meaning of both of these words, and how they are related but not equivalent.

First, we will give the formal definition of independence:

Definition (
Independence of Random Variables)
.

Two random variables
X
X
X
and
Y
Y
Y
are
independent
if the joint probability distribution
P
(
X
,
Y
)
P(X, Y)
P
(
X
,
Y
)
can be written as the product of the two individual distributions. That is,

Essentially this means that the joint probability of the random variables
X
X
X
and
Y
Y
Y
together are actually separable into the product of their individual probabilities. Here are some other equivalent definitions:

This first alternative definition states that the probability of any outcome of
X
X
X
and any outcome of
Y
Y
Y
occurring simultaneously is the product of those individual probabilities.